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This account of basic manifold theory and global analysis, based on senior undergraduate and postgraduate courses at Glasgow for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology, the latter of which is included in two appendices, as many courses on mathematics for physics students do not include this subject. Topics covered include vector spaces; tensor algebra; differentiable manifolds; exterior differential forms; pseudo-Riemannian and Riemannian manifolds; sympletic manifolds; and complex linear algebra.
1 Vector Spaces
2 Tensor Algebra
3 Differentiable Manifolds
4 Vector and Tensor Fields on a Manifold
5 Exterior Differential Forms
6 Differentiation on a Manifold
7 Pseudo-Riemannian and Riemannian Manifolds
8 Symplectic Manifolds
9 Lie Groups
10 Integration on a Manifold
11 Fibre Bundles
12 Complex Linear Algebra, Almost Complex Manifolds
Appendix 1: Analytic Topology
Appendix 2: Quaternions and Cayley Numbers
Appendix 3: The Semidirect Product of Two Groups
Appendix 4: Homotopy Review
Some Answers, Some Hints and Som Fragmentary Solutions to the Exercises
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